Cristian M. answered • 08/06/20

UC Davis Grad Offers Patient, Explorative, and Clear Math Tutoring

**Question**: Gilbert vacuums the rugs every 14 days; checks on his neighbor’s cat every 2 days; and pays his rent every 30 days. Today he did all three. How long will it be before he has another day like today? Show how.

**Answer**: Let's find the least common multiple (LCM) of these numbers.

Prime factorizations

-----------------------------

(don't worry about factoring out 1 from each number)

(vacuuming) 14: 2 * 7

(cat) 2: 2

(rent) 30: 2 * 3 * 5

-----------------------------

Look at all the individual prime factors that we have. We have 2, 3, 5, and 7 as unique prime factors.

-Who has the most of the factor 2? All of the figures have one copy, so include 2 one time in your answer.

-Who has the most of the factor 3? The rent figure has the only copy of it, so include 3 one time in your answer.

-Who has the most of the factor 5? The rent figure has the only copy of it, so include 5 one time in your answer.

-Who has the most of the factor 7? The vacuuming figure has the only copy of it, so include 7 one time in your answer.

(So your problem was easy in terms of number of copies of factors. If you have a number like 8, for example, it factors to 2*2*2, which, compared to 2, 3, 5, and 7, has the most copies of the unique prime factor of 2. It would be included three times in the final answer.)

The copies we saved need to be multiplied together:

2 * 3 * 5 * 7 = **210**

**--------------------------------------**

Here's another way to look at it, more for the sake of visual intuition:

*every 14 days*: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, **210**

*every 2 days*: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, **210**

*every 30 days*: 30, 60, 90, 120, 150, 180, **210**

This is why we find the LCM of the three numbers, precisely to save us all of that work to find the nearest common ground. That's what the LCM helps us to find, the earliest meeting place that works for everyone. Work smarter, not harder.

Gilbert will have another day like this, with all three tasks, **210 days** from now.